Calculus Examples

$\frac{d y}{d x} + y = 3 e^{2 x}$

Step 1

The integrating factor is defined by the formula $e^{\int P (x) d x}$ , where $P (x) = 1$ .

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Step 1.1

Set up the integration.

$e^{\int d x}$

Step 1.2

Apply the constant rule.

$e^{x + C}$

Step 1.3

Remove the constant of integration.

$e^{x}$

Step 2

Multiply each term by the integrating factor $e^{x}$ .

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Step 2.1

Multiply each term by $e^{x}$ .

$e^{x} \frac{d y}{d x} + e^{x} y = e^{x} (3 e^{2 x})$

Step 2.2

Rewrite using the commutative property of multiplication.

$e^{x} \frac{d y}{d x} + e^{x} y = 3 e^{x} e^{2 x}$

Step 2.3

Multiply $e^{x}$ by $e^{2 x}$ by adding the exponents.

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Step 2.3.1

Move $e^{2 x}$ .

$e^{x} \frac{d y}{d x} + e^{x} y = 3 (e^{2 x} e^{x})$

Step 2.3.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$e^{x} \frac{d y}{d x} + e^{x} y = 3 e^{2 x + x}$

Step 2.3.3

Add $2 x$ and $x$ .

$e^{x} \frac{d y}{d x} + e^{x} y = 3 e^{3 x}$

Step 2.4

Reorder factors in $e^{x} \frac{d y}{d x} + e^{x} y = 3 e^{3 x}$ .

$e^{x} \frac{d y}{d x} + y e^{x} = 3 e^{3 x}$

Step 3

Rewrite the left side as a result of differentiating a product.

$\frac{d}{d x} [e^{x} y] = 3 e^{3 x}$

Step 4

Set up an integral on each side.

$\int \frac{d}{d x} [e^{x} y] d x = \int 3 e^{3 x} d x$

Step 5

Integrate the left side.

$e^{x} y = \int 3 e^{3 x} d x$

Step 6

Integrate the right side.

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Step 6.1

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$e^{x} y = 3 \int e^{3 x} d x$

Step 6.2

Let $u = 3 x$ . Then $d u = 3 d x$ , so $\frac{1}{3} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 6.2.1

Let $u = 3 x$ . Find $\frac{d u}{d x}$ .

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Step 6.2.1.1

Differentiate $3 x$ .

$\frac{d}{d x} [3 x]$

Step 6.2.1.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$3 \frac{d}{d x} [x]$

Step 6.2.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$3 \cdot 1$

Step 6.2.1.4

Multiply $3$ by $1$ .

$3$

Step 6.2.2

Rewrite the problem using $u$ and $d u$ .

$e^{x} y = 3 \int e^{u} \frac{1}{3} d u$

Step 6.3

Combine $e^{u}$ and $\frac{1}{3}$ .

$e^{x} y = 3 \int \frac{e^{u}}{3} d u$

Step 6.4

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

$e^{x} y = 3 (\frac{1}{3} \int e^{u} d u)$

Step 6.5

Simplify.

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Step 6.5.1

Combine $\frac{1}{3}$ and $3$ .

$e^{x} y = \frac{3}{3} \int e^{u} d u$

Step 6.5.2

Cancel the common factor of $3$ .

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Step 6.5.2.1

Cancel the common factor.

$e^{x} y = \frac{3}{3} \int e^{u} d u$

Step 6.5.2.2

Rewrite the expression.

$e^{x} y = 1 \int e^{u} d u$

Step 6.5.3

Multiply $\int e^{u} d u$ by $1$ .

$e^{x} y = \int e^{u} d u$

Step 6.6

The integral of $e^{u}$ with respect to $u$ is $e^{u}$ .

$e^{x} y = e^{u} + C$

Step 6.7

Replace all occurrences of $u$ with $3 x$ .

$e^{x} y = e^{3 x} + C$

Step 7

Divide each term in $e^{x} y = e^{3 x} + C$ by $e^{x}$ and simplify.

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Step 7.1

Divide each term in $e^{x} y = e^{3 x} + C$ by $e^{x}$ .

$\frac{e^{x} y}{e^{x}} = \frac{e^{3 x}}{e^{x}} + \frac{C}{e^{x}}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $e^{x}$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{e^{x} y}{e^{x}} = \frac{e^{3 x}}{e^{x}} + \frac{C}{e^{x}}$

Step 7.2.1.2

Divide $y$ by $1$ .

$y = \frac{e^{3 x}}{e^{x}} + \frac{C}{e^{x}}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Cancel the common factor of $e^{3 x}$ and $e^{x}$ .

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Step 7.3.1.1

Factor $e^{x}$ out of $e^{3 x}$ .

$y = \frac{e^{x} e^{2 x}}{e^{x}} + \frac{C}{e^{x}}$

Step 7.3.1.2

Cancel the common factors.

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Step 7.3.1.2.1

Multiply by $1$ .

$y = \frac{e^{x} e^{2 x}}{e^{x} \cdot 1} + \frac{C}{e^{x}}$

Step 7.3.1.2.2

Cancel the common factor.

$y = \frac{e^{x} e^{2 x}}{e^{x} \cdot 1} + \frac{C}{e^{x}}$

Step 7.3.1.2.3

Rewrite the expression.

$y = \frac{e^{2 x}}{1} + \frac{C}{e^{x}}$

Step 7.3.1.2.4

Divide $e^{2 x}$ by $1$ .

$y = e^{2 x} + \frac{C}{e^{x}}$